A neighborhood is just an open set. An open ball requires being in a metric space. The word neighborhood is usually used as opposed to just open set because you want to give the impression that the open set is supposed to be a small one, similar to saying let [tex] \epsilon>0[/tex] vs saying let [tex]M>0[/tex]. They both say the exact same thing but one of them indicates we're interested in picking small numbers and one large numbers. It's not a formal definition but just to give the reader some intuition